This book presents a hierarchy of macroscopic models for semiconductor devices, studying three classes of models in detail: isentropic drift-diffusion equations, energy-transport models, and quantum hydrodynamic equations. The derivation of each, including physical discussions, is shown. Numerical simulations for modern semiconductor devices are performed, showing the particular features of each. The author develops modern analytical techniques, such as positive solution methods, local energy methods for free-boundary problems and entropy methods.
1 Introduction.- 1.1 A hierarchy of semiconductor models.- 1.2 Quasi-hydrodynamic semiconductor models.- 2 Basic Semiconductor Physics.- 2.1 Homogeneous semiconductors.- 2.2 Inhomogeneous semiconductors.- 3 The Isentropic Drift-diffusion Model.- 3.1 Derivation of the model.- 3.1.1 Semiconductor equations based on Fermi-Dirac statistics.- 3.1.2 The isentropic model-scaling.- 3.1.3 The convergence result.- 3.2 Existence of transient solutions.- 3.2.1 Assumptions and existence result.- 3.2.2 Proof of the existence result.- 3.3 Uniqueness of transient solutions.- 3.4 Localization of vacuum solutions.- 3.4.1 Main results.- 3.4.2 Proofs of the main results.- 3.4.3 Numerical examples.- 3.5 Numerical approximation.- 3.5.1 The mixed finite element discretization in one space dimension.- 3.5.2 Numerical examples in one space dimension.- 3.5.3 The mixed finite element discretization in two space dimensions.- 3.5.4 Numerical examples in two space dimensions.- 3.6 Current-voltage characteristics.- 3.6.1 Numerical current-voltage characteristics.- 3.6.2 High-injection current-voltage characteristics.- 4 The Energy-transport Model.- 4.1 Derivation of the model.- 4.1.1 General non-parabolic band diagrams.- 4.1.2 A drift-diffusion formulation for the current densities.- 4.1.3 A non-parabolic band approximation.- 4.1.4 Parabolic band approximation.- 4.2 Symmetrization and entropy function.- 4.3 Existence of transient solutions.- 4.3.1 Assumptions and main results.- 4.3.2 Semidiscretization.- 4.3.3 Proof of the existence result.- 4.4 Long-time behavior of the transient solution.- 4.5 Regularity and uniqueness.- 4.5.1 Regularity of transient solutions.- 4.5.2 Uniqueness of transient solutions.- 4.6 Existence of steady-state solutions.- 4.7 Uniqueness of steady-state solutions.- 4.8 Numerical approximation.- 4.8.1 The mixed finite element discretization in one space dimension.- 4.8.2 Numerical results.- 5 The Quantum Hydrodynamic Model.- 5.1 Derivation of the model.- 5.2 Existence and positivity.- 5.2.1 Existence of steady-state solutions.- 5.2.2 Positivity and non-positivity properties.- 5.3 Uniqueness of steady-state solutions.- 5.4 A non-existence result.- 5.5 The classical limit.- 5.5.1 The classical limit of the thermal equilibrium state.- 5.5.2 The classical limit in the `subsonic' steady state.- 5.5.3 Numerical examples.- 5.6 Current-voltage characteristics.- 5.6.1 Scaling of the equations.- 5.6.2 Analytical and numerical current-voltage characteristics.- 5.7 A positivity-preserving numerical scheme.- 5.7.1 Semidiscretization in time.- 5.7.2 Stability bounds and convergence results.- 5.7.3 Numerical examples.- References.
